A note on arithmetic-geometric-harmonic mean inequality of several positive operators

Document Type : Research Article

Authors

1 Department of Mathematics, Faculty of science, University of Zanjan, Zanjan, Iran.

2 Department of Mathematics, Faculty of science, University of Zanjan, Zanjan, Iran

Abstract

‎Suppose that $B_1,\cdots,B_m$ are positive operators on a Hilbert space $\mathcal{H}$‎. ‎In this paper we generalize the weighted arithmetic‎, ‎geometric and harmonic means as follows‎:

‎\begin{align*}‎

‎{\mathbf a_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf a_2}(k_1,N';B_1,{\mathbf a_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\frac{k_1B_1+\cdots+k_mB_m}{N}\\‎

‎{\mathbf h_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf h_2}(k_1,N';B_1,{\mathbf h_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\left(\frac{k_1B_1^{-1}+\cdots+k_mB_m^{-1}}{N}\right)^{-1}\\‎

‎{\mathbf g_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf g_2}(k_1,N';B_1,{\mathbf g_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))‎

‎\end{align*}‎

‎where $\boldsymbol\kappa=(k_1,\cdots,k_m)‎, ‎N=k_1+\cdots+k_m‎, ‎\boldsymbol\kappa'=(k_2,\cdots‎, ‎k_m)$ and $N'=k_2+\cdots+k_m$‎. ‎We show that the arithmetic-geometric-harmonic mean inequality holds‎. ‎Also we investigate nine property of the geometric mean‎.

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References

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Journal of the Iranian Mathematical Society
Volume 5, Issue 1
January 2024
Pages 33-43

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